$ A = \left[\begin{array}{rrr}3 & 2 & 2 \\ 1 & 1 & -1\end{array}\right]$ $ C = \left[\begin{array}{rr}0 & 1 \\ 0 & -1 \\ 1 & 0\end{array}\right]$ What is $ A C$ ?
Explanation: Because $ A$ has dimensions $(2\times3)$ and $ C$ has dimensions $(3\times2)$ , the answer matrix will have dimensions $(2\times2)$ $ A C = \left[\begin{array}{rrr}{3} & {2} & {2} \\ {1} & {1} & {-1}\end{array}\right] \left[\begin{array}{rr}{0} & \color{#DF0030}{1} \\ {0} & \color{#DF0030}{-1} \\ {1} & \color{#DF0030}{0}\end{array}\right] = \left[\begin{array}{rr}? & ? \\ ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ C$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ C$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ C$ , and so on. Add the products together. $ \left[\begin{array}{rr}{3}\cdot{0}+{2}\cdot{0}+{2}\cdot{1} & ? \\ ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{0}+{2}\cdot{0}+{2}\cdot{1} & ? \\ {1}\cdot{0}+{1}\cdot{0}+{-1}\cdot{1} & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ C$ and add the products together. $ \left[\begin{array}{rr}{3}\cdot{0}+{2}\cdot{0}+{2}\cdot{1} & {3}\cdot\color{#DF0030}{1}+{2}\cdot\color{#DF0030}{-1}+{2}\cdot\color{#DF0030}{0} \\ {1}\cdot{0}+{1}\cdot{0}+{-1}\cdot{1} & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rr}{3}\cdot{0}+{2}\cdot{0}+{2}\cdot{1} & {3}\cdot\color{#DF0030}{1}+{2}\cdot\color{#DF0030}{-1}+{2}\cdot\color{#DF0030}{0} \\ {1}\cdot{0}+{1}\cdot{0}+{-1}\cdot{1} & {1}\cdot\color{#DF0030}{1}+{1}\cdot\color{#DF0030}{-1}+{-1}\cdot\color{#DF0030}{0}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rr}2 & 1 \\ -1 & 0\end{array}\right] $